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Theorem cocanfo 36582
Description: Cancellation of a surjective function from the right side of a composition. (Contributed by Jeff Madsen, 1-Jun-2011.) (Proof shortened by Mario Carneiro, 27-Dec-2014.)
Assertion
Ref Expression
cocanfo (((𝐹:𝐴onto𝐵𝐺 Fn 𝐵𝐻 Fn 𝐵) ∧ (𝐺𝐹) = (𝐻𝐹)) → 𝐺 = 𝐻)

Proof of Theorem cocanfo
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simplr 767 . . . . . 6 ((((𝐹:𝐴onto𝐵𝐺 Fn 𝐵𝐻 Fn 𝐵) ∧ (𝐺𝐹) = (𝐻𝐹)) ∧ 𝑦𝐴) → (𝐺𝐹) = (𝐻𝐹))
21fveq1d 6893 . . . . 5 ((((𝐹:𝐴onto𝐵𝐺 Fn 𝐵𝐻 Fn 𝐵) ∧ (𝐺𝐹) = (𝐻𝐹)) ∧ 𝑦𝐴) → ((𝐺𝐹)‘𝑦) = ((𝐻𝐹)‘𝑦))
3 simpl1 1191 . . . . . . 7 (((𝐹:𝐴onto𝐵𝐺 Fn 𝐵𝐻 Fn 𝐵) ∧ (𝐺𝐹) = (𝐻𝐹)) → 𝐹:𝐴onto𝐵)
4 fof 6805 . . . . . . 7 (𝐹:𝐴onto𝐵𝐹:𝐴𝐵)
53, 4syl 17 . . . . . 6 (((𝐹:𝐴onto𝐵𝐺 Fn 𝐵𝐻 Fn 𝐵) ∧ (𝐺𝐹) = (𝐻𝐹)) → 𝐹:𝐴𝐵)
6 fvco3 6990 . . . . . 6 ((𝐹:𝐴𝐵𝑦𝐴) → ((𝐺𝐹)‘𝑦) = (𝐺‘(𝐹𝑦)))
75, 6sylan 580 . . . . 5 ((((𝐹:𝐴onto𝐵𝐺 Fn 𝐵𝐻 Fn 𝐵) ∧ (𝐺𝐹) = (𝐻𝐹)) ∧ 𝑦𝐴) → ((𝐺𝐹)‘𝑦) = (𝐺‘(𝐹𝑦)))
8 fvco3 6990 . . . . . 6 ((𝐹:𝐴𝐵𝑦𝐴) → ((𝐻𝐹)‘𝑦) = (𝐻‘(𝐹𝑦)))
95, 8sylan 580 . . . . 5 ((((𝐹:𝐴onto𝐵𝐺 Fn 𝐵𝐻 Fn 𝐵) ∧ (𝐺𝐹) = (𝐻𝐹)) ∧ 𝑦𝐴) → ((𝐻𝐹)‘𝑦) = (𝐻‘(𝐹𝑦)))
102, 7, 93eqtr3d 2780 . . . 4 ((((𝐹:𝐴onto𝐵𝐺 Fn 𝐵𝐻 Fn 𝐵) ∧ (𝐺𝐹) = (𝐻𝐹)) ∧ 𝑦𝐴) → (𝐺‘(𝐹𝑦)) = (𝐻‘(𝐹𝑦)))
1110ralrimiva 3146 . . 3 (((𝐹:𝐴onto𝐵𝐺 Fn 𝐵𝐻 Fn 𝐵) ∧ (𝐺𝐹) = (𝐻𝐹)) → ∀𝑦𝐴 (𝐺‘(𝐹𝑦)) = (𝐻‘(𝐹𝑦)))
12 fveq2 6891 . . . . . 6 ((𝐹𝑦) = 𝑥 → (𝐺‘(𝐹𝑦)) = (𝐺𝑥))
13 fveq2 6891 . . . . . 6 ((𝐹𝑦) = 𝑥 → (𝐻‘(𝐹𝑦)) = (𝐻𝑥))
1412, 13eqeq12d 2748 . . . . 5 ((𝐹𝑦) = 𝑥 → ((𝐺‘(𝐹𝑦)) = (𝐻‘(𝐹𝑦)) ↔ (𝐺𝑥) = (𝐻𝑥)))
1514cbvfo 7286 . . . 4 (𝐹:𝐴onto𝐵 → (∀𝑦𝐴 (𝐺‘(𝐹𝑦)) = (𝐻‘(𝐹𝑦)) ↔ ∀𝑥𝐵 (𝐺𝑥) = (𝐻𝑥)))
163, 15syl 17 . . 3 (((𝐹:𝐴onto𝐵𝐺 Fn 𝐵𝐻 Fn 𝐵) ∧ (𝐺𝐹) = (𝐻𝐹)) → (∀𝑦𝐴 (𝐺‘(𝐹𝑦)) = (𝐻‘(𝐹𝑦)) ↔ ∀𝑥𝐵 (𝐺𝑥) = (𝐻𝑥)))
1711, 16mpbid 231 . 2 (((𝐹:𝐴onto𝐵𝐺 Fn 𝐵𝐻 Fn 𝐵) ∧ (𝐺𝐹) = (𝐻𝐹)) → ∀𝑥𝐵 (𝐺𝑥) = (𝐻𝑥))
18 eqfnfv 7032 . . . 4 ((𝐺 Fn 𝐵𝐻 Fn 𝐵) → (𝐺 = 𝐻 ↔ ∀𝑥𝐵 (𝐺𝑥) = (𝐻𝑥)))
19183adant1 1130 . . 3 ((𝐹:𝐴onto𝐵𝐺 Fn 𝐵𝐻 Fn 𝐵) → (𝐺 = 𝐻 ↔ ∀𝑥𝐵 (𝐺𝑥) = (𝐻𝑥)))
2019adantr 481 . 2 (((𝐹:𝐴onto𝐵𝐺 Fn 𝐵𝐻 Fn 𝐵) ∧ (𝐺𝐹) = (𝐻𝐹)) → (𝐺 = 𝐻 ↔ ∀𝑥𝐵 (𝐺𝑥) = (𝐻𝑥)))
2117, 20mpbird 256 1 (((𝐹:𝐴onto𝐵𝐺 Fn 𝐵𝐻 Fn 𝐵) ∧ (𝐺𝐹) = (𝐻𝐹)) → 𝐺 = 𝐻)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wral 3061  ccom 5680   Fn wfn 6538  wf 6539  ontowfo 6541  cfv 6543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fo 6549  df-fv 6551
This theorem is referenced by: (None)
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